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1 747 jeremybenn 
// Copyright 2010 The Go Authors. All rights reserved.
2 
// Use of this source code is governed by a BSD-style
3 
// license that can be found in the LICENSE file.
4 
 
5 
package math
6 
 
7 
/*
8 
        Bessel function of the first and second kinds of order one.
9 
*/
10 
 
11 
// The original C code and the long comment below are
12 
// from FreeBSD's /usr/src/lib/msun/src/e_j1.c and
13 
// came with this notice.  The go code is a simplified
14 
// version of the original C.
15 
//
16 
// ====================================================
17 
// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
18 
//
19 
// Developed at SunPro, a Sun Microsystems, Inc. business.
20 
// Permission to use, copy, modify, and distribute this
21 
// software is freely granted, provided that this notice
22 
// is preserved.
23 
// ====================================================
24 
//
25 
// __ieee754_j1(x), __ieee754_y1(x)
26 
// Bessel function of the first and second kinds of order one.
27 
// Method -- j1(x):
28 
//      1. For tiny x, we use j1(x) = x/2 - x**3/16 + x**5/384 - ...
29 
//      2. Reduce x to |x| since j1(x)=-j1(-x),  and
30 
//         for x in (0,2)
31 
//              j1(x) = x/2 + x*z*R0/S0,  where z = x*x;
32 
//         (precision:  |j1/x - 1/2 - R0/S0 |<2**-61.51 )
33 
//         for x in (2,inf)
34 
//              j1(x) = sqrt(2/(pi*x))*(p1(x)*cos(x1)-q1(x)*sin(x1))
35 
//              y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
36 
//         where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
37 
//         as follow:
38 
//              cos(x1) =  cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
39 
//                      =  1/sqrt(2) * (sin(x) - cos(x))
40 
//              sin(x1) =  sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
41 
//                      = -1/sqrt(2) * (sin(x) + cos(x))
42 
//         (To avoid cancellation, use
43 
//              sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
44 
//         to compute the worse one.)
45 
//
46 
//      3 Special cases
47 
//              j1(nan)= nan
48 
//              j1(0) = 0
49 
//              j1(inf) = 0
50 
//
51 
// Method -- y1(x):
52 
//      1. screen out x<=0 cases: y1(0)=-inf, y1(x<0)=NaN
53 
//      2. For x<2.
54 
//         Since
55 
//              y1(x) = 2/pi*(j1(x)*(ln(x/2)+Euler)-1/x-x/2+5/64*x**3-...)
56 
//         therefore y1(x)-2/pi*j1(x)*ln(x)-1/x is an odd function.
57 
//         We use the following function to approximate y1,
58 
//              y1(x) = x*U(z)/V(z) + (2/pi)*(j1(x)*ln(x)-1/x), z= x**2
59 
//         where for x in [0,2] (abs err less than 2**-65.89)
60 
//              U(z) = U0[0] + U0[1]*z + ... + U0[4]*z**4
61 
//              V(z) = 1  + v0[0]*z + ... + v0[4]*z**5
62 
//         Note: For tiny x, 1/x dominate y1 and hence
63 
//              y1(tiny) = -2/pi/tiny, (choose tiny<2**-54)
64 
//      3. For x>=2.
65 
//               y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x1)+q1(x)*cos(x1))
66 
//         where x1 = x-3*pi/4. It is better to compute sin(x1),cos(x1)
67 
//         by method mentioned above.
68 
 
69 
// J1 returns the order-one Bessel function of the first kind.
70 
//
71 
// Special cases are:
72 
//      J1(±Inf) = 0
73 
//      J1(NaN) = NaN
74 
func J1(x float64) float64 {
75 
        const (
76 
                TwoM27 = 1.0 / (1 << 27) // 2**-27 0x3e40000000000000
77 
                Two129 = 1 << 129        // 2**129 0x4800000000000000
78 
                // R0/S0 on [0, 2]
79 
                R00 = -6.25000000000000000000e-02 // 0xBFB0000000000000
80 
                R01 = 1.40705666955189706048e-03  // 0x3F570D9F98472C61
81 
                R02 = -1.59955631084035597520e-05 // 0xBEF0C5C6BA169668
82 
                R03 = 4.96727999609584448412e-08  // 0x3E6AAAFA46CA0BD9
83 
                S01 = 1.91537599538363460805e-02  // 0x3F939D0B12637E53
84 
                S02 = 1.85946785588630915560e-04  // 0x3F285F56B9CDF664
85 
                S03 = 1.17718464042623683263e-06  // 0x3EB3BFF8333F8498
86 
                S04 = 5.04636257076217042715e-09  // 0x3E35AC88C97DFF2C
87 
                S05 = 1.23542274426137913908e-11  // 0x3DAB2ACFCFB97ED8
88 
        )
89 
        // special cases
90 
        switch {
91 
        case IsNaN(x):
92 
                return x
93 
        case IsInf(x, 0) || x == 0:
94 
                return 0
95 
        }
96 
 
97 
        sign := false
98 
        if x < 0 {
99 
                x = -x
100 
                sign = true
101 
        }
102 
        if x >= 2 {
103 
                s, c := Sincos(x)
104 
                ss := -s - c
105 
                cc := s - c
106 
 
107 
                // make sure x+x does not overflow
108 
                if x < MaxFloat64/2 {
109 
                        z := Cos(x + x)
110 
                        if s*c > 0 {
111 
                                cc = z / ss
112 
                        } else {
113 
                                ss = z / cc
114 
                        }
115 
                }
116 
 
117 
                // j1(x) = 1/sqrt(pi) * (P(1,x)*cc - Q(1,x)*ss) / sqrt(x)
118 
                // y1(x) = 1/sqrt(pi) * (P(1,x)*ss + Q(1,x)*cc) / sqrt(x)
119 
 
120 
                var z float64
121 
                if x > Two129 {
122 
                        z = (1 / SqrtPi) * cc / Sqrt(x)
123 
                } else {
124 
                        u := pone(x)
125 
                        v := qone(x)
126 
                        z = (1 / SqrtPi) * (u*cc - v*ss) / Sqrt(x)
127 
                }
128 
                if sign {
129 
                        return -z
130 
                }
131 
                return z
132 
        }
133 
        if x < TwoM27 { // |x|<2**-27
134 
                return 0.5 * x // inexact if x!=0 necessary
135 
        }
136 
        z := x * x
137 
        r := z * (R00 + z*(R01+z*(R02+z*R03)))
138 
        s := 1.0 + z*(S01+z*(S02+z*(S03+z*(S04+z*S05))))
139 
        r *= x
140 
        z = 0.5*x + r/s
141 
        if sign {
142 
                return -z
143 
        }
144 
        return z
145 
}
146 
 
147 
// Y1 returns the order-one Bessel function of the second kind.
148 
//
149 
// Special cases are:
150 
//      Y1(+Inf) = 0
151 
//      Y1(0) = -Inf
152 
//      Y1(x < 0) = NaN
153 
//      Y1(NaN) = NaN
154 
func Y1(x float64) float64 {
155 
        const (
156 
                TwoM54 = 1.0 / (1 << 54)             // 2**-54 0x3c90000000000000
157 
                Two129 = 1 << 129                    // 2**129 0x4800000000000000
158 
                U00    = -1.96057090646238940668e-01 // 0xBFC91866143CBC8A
159 
                U01    = 5.04438716639811282616e-02  // 0x3FA9D3C776292CD1
160 
                U02    = -1.91256895875763547298e-03 // 0xBF5F55E54844F50F
161 
                U03    = 2.35252600561610495928e-05  // 0x3EF8AB038FA6B88E
162 
                U04    = -9.19099158039878874504e-08 // 0xBE78AC00569105B8
163 
                V00    = 1.99167318236649903973e-02  // 0x3F94650D3F4DA9F0
164 
                V01    = 2.02552581025135171496e-04  // 0x3F2A8C896C257764
165 
                V02    = 1.35608801097516229404e-06  // 0x3EB6C05A894E8CA6
166 
                V03    = 6.22741452364621501295e-09  // 0x3E3ABF1D5BA69A86
167 
                V04    = 1.66559246207992079114e-11  // 0x3DB25039DACA772A
168 
        )
169 
        // special cases
170 
        switch {
171 
        case x < 0 || IsNaN(x):
172 
                return NaN()
173 
        case IsInf(x, 1):
174 
                return 0
175 
        case x == 0:
176 
                return Inf(-1)
177 
        }
178 
 
179 
        if x >= 2 {
180 
                s, c := Sincos(x)
181 
                ss := -s - c
182 
                cc := s - c
183 
 
184 
                // make sure x+x does not overflow
185 
                if x < MaxFloat64/2 {
186 
                        z := Cos(x + x)
187 
                        if s*c > 0 {
188 
                                cc = z / ss
189 
                        } else {
190 
                                ss = z / cc
191 
                        }
192 
                }
193 
                // y1(x) = sqrt(2/(pi*x))*(p1(x)*sin(x0)+q1(x)*cos(x0))
194 
                // where x0 = x-3pi/4
195 
                //     Better formula:
196 
                //         cos(x0) = cos(x)cos(3pi/4)+sin(x)sin(3pi/4)
197 
                //                 =  1/sqrt(2) * (sin(x) - cos(x))
198 
                //         sin(x0) = sin(x)cos(3pi/4)-cos(x)sin(3pi/4)
199 
                //                 = -1/sqrt(2) * (cos(x) + sin(x))
200 
                // To avoid cancellation, use
201 
                //     sin(x) +- cos(x) = -cos(2x)/(sin(x) -+ cos(x))
202 
                // to compute the worse one.
203 
 
204 
                var z float64
205 
                if x > Two129 {
206 
                        z = (1 / SqrtPi) * ss / Sqrt(x)
207 
                } else {
208 
                        u := pone(x)
209 
                        v := qone(x)
210 
                        z = (1 / SqrtPi) * (u*ss + v*cc) / Sqrt(x)
211 
                }
212 
                return z
213 
        }
214 
        if x <= TwoM54 { // x < 2**-54
215 
                return -(2 / Pi) / x
216 
        }
217 
        z := x * x
218 
        u := U00 + z*(U01+z*(U02+z*(U03+z*U04)))
219 
        v := 1 + z*(V00+z*(V01+z*(V02+z*(V03+z*V04))))
220 
        return x*(u/v) + (2/Pi)*(J1(x)*Log(x)-1/x)
221 
}
222 
 
223 
// For x >= 8, the asymptotic expansions of pone is
224 
//      1 + 15/128 s**2 - 4725/2**15 s**4 - ..., where s = 1/x.
225 
// We approximate pone by
226 
//      pone(x) = 1 + (R/S)
227 
// where R = pr0 + pr1*s**2 + pr2*s**4 + ... + pr5*s**10
228 
//       S = 1 + ps0*s**2 + ... + ps4*s**10
229 
// and
230 
//      | pone(x)-1-R/S | <= 2**(-60.06)
231 
 
232 
// for x in [inf, 8]=1/[0,0.125]
233 
var p1R8 = [6]float64{
234 
        0.00000000000000000000e+00, // 0x0000000000000000
235 
        1.17187499999988647970e-01, // 0x3FBDFFFFFFFFFCCE
236 
        1.32394806593073575129e+01, // 0x402A7A9D357F7FCE
237 
        4.12051854307378562225e+02, // 0x4079C0D4652EA590
238 
        3.87474538913960532227e+03, // 0x40AE457DA3A532CC
239 
        7.91447954031891731574e+03, // 0x40BEEA7AC32782DD
240 
}
241 
var p1S8 = [5]float64{
242 
        1.14207370375678408436e+02, // 0x405C8D458E656CAC
243 
        3.65093083420853463394e+03, // 0x40AC85DC964D274F
244 
        3.69562060269033463555e+04, // 0x40E20B8697C5BB7F
245 
        9.76027935934950801311e+04, // 0x40F7D42CB28F17BB
246 
        3.08042720627888811578e+04, // 0x40DE1511697A0B2D
247 
}
248 
 
249 
// for x in [8,4.5454] = 1/[0.125,0.22001]
250 
var p1R5 = [6]float64{
251 
        1.31990519556243522749e-11, // 0x3DAD0667DAE1CA7D
252 
        1.17187493190614097638e-01, // 0x3FBDFFFFE2C10043
253 
        6.80275127868432871736e+00, // 0x401B36046E6315E3
254 
        1.08308182990189109773e+02, // 0x405B13B9452602ED
255 
        5.17636139533199752805e+02, // 0x40802D16D052D649
256 
        5.28715201363337541807e+02, // 0x408085B8BB7E0CB7
257 
}
258 
var p1S5 = [5]float64{
259 
        5.92805987221131331921e+01, // 0x404DA3EAA8AF633D
260 
        9.91401418733614377743e+02, // 0x408EFB361B066701
261 
        5.35326695291487976647e+03, // 0x40B4E9445706B6FB
262 
        7.84469031749551231769e+03, // 0x40BEA4B0B8A5BB15
263 
        1.50404688810361062679e+03, // 0x40978030036F5E51
264 
}
265 
 
266 
// for x in[4.5453,2.8571] = 1/[0.2199,0.35001]
267 
var p1R3 = [6]float64{
268 
        3.02503916137373618024e-09, // 0x3E29FC21A7AD9EDD
269 
        1.17186865567253592491e-01, // 0x3FBDFFF55B21D17B
270 
        3.93297750033315640650e+00, // 0x400F76BCE85EAD8A
271 
        3.51194035591636932736e+01, // 0x40418F489DA6D129
272 
        9.10550110750781271918e+01, // 0x4056C3854D2C1837
273 
        4.85590685197364919645e+01, // 0x4048478F8EA83EE5
274 
}
275 
var p1S3 = [5]float64{
276 
        3.47913095001251519989e+01, // 0x40416549A134069C
277 
        3.36762458747825746741e+02, // 0x40750C3307F1A75F
278 
        1.04687139975775130551e+03, // 0x40905B7C5037D523
279 
        8.90811346398256432622e+02, // 0x408BD67DA32E31E9
280 
        1.03787932439639277504e+02, // 0x4059F26D7C2EED53
281 
}
282 
 
283 
// for x in [2.8570,2] = 1/[0.3499,0.5]
284 
var p1R2 = [6]float64{
285 
        1.07710830106873743082e-07, // 0x3E7CE9D4F65544F4
286 
        1.17176219462683348094e-01, // 0x3FBDFF42BE760D83
287 
        2.36851496667608785174e+00, // 0x4002F2B7F98FAEC0
288 
        1.22426109148261232917e+01, // 0x40287C377F71A964
289 
        1.76939711271687727390e+01, // 0x4031B1A8177F8EE2
290 
        5.07352312588818499250e+00, // 0x40144B49A574C1FE
291 
}
292 
var p1S2 = [5]float64{
293 
        2.14364859363821409488e+01, // 0x40356FBD8AD5ECDC
294 
        1.25290227168402751090e+02, // 0x405F529314F92CD5
295 
        2.32276469057162813669e+02, // 0x406D08D8D5A2DBD9
296 
        1.17679373287147100768e+02, // 0x405D6B7ADA1884A9
297 
        8.36463893371618283368e+00, // 0x4020BAB1F44E5192
298 
}
299 
 
300 
func pone(x float64) float64 {
301 
        var p [6]float64
302 
        var q [5]float64
303 
        if x >= 8 {
304 
                p = p1R8
305 
                q = p1S8
306 
        } else if x >= 4.5454 {
307 
                p = p1R5
308 
                q = p1S5
309 
        } else if x >= 2.8571 {
310 
                p = p1R3
311 
                q = p1S3
312 
        } else if x >= 2 {
313 
                p = p1R2
314 
                q = p1S2
315 
        }
316 
        z := 1 / (x * x)
317 
        r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
318 
        s := 1.0 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*q[4]))))
319 
        return 1 + r/s
320 
}
321 
 
322 
// For x >= 8, the asymptotic expansions of qone is
323 
//      3/8 s - 105/1024 s**3 - ..., where s = 1/x.
324 
// We approximate qone by
325 
//      qone(x) = s*(0.375 + (R/S))
326 
// where R = qr1*s**2 + qr2*s**4 + ... + qr5*s**10
327 
//       S = 1 + qs1*s**2 + ... + qs6*s**12
328 
// and
329 
//      | qone(x)/s -0.375-R/S | <= 2**(-61.13)
330 
 
331 
// for x in [inf, 8] = 1/[0,0.125]
332 
var q1R8 = [6]float64{
333 
        0.00000000000000000000e+00,  // 0x0000000000000000
334 
        -1.02539062499992714161e-01, // 0xBFBA3FFFFFFFFDF3
335 
        -1.62717534544589987888e+01, // 0xC0304591A26779F7
336 
        -7.59601722513950107896e+02, // 0xC087BCD053E4B576
337 
        -1.18498066702429587167e+04, // 0xC0C724E740F87415
338 
        -4.84385124285750353010e+04, // 0xC0E7A6D065D09C6A
339 
}
340 
var q1S8 = [6]float64{
341 
        1.61395369700722909556e+02,  // 0x40642CA6DE5BCDE5
342 
        7.82538599923348465381e+03,  // 0x40BE9162D0D88419
343 
        1.33875336287249578163e+05,  // 0x4100579AB0B75E98
344 
        7.19657723683240939863e+05,  // 0x4125F65372869C19
345 
        6.66601232617776375264e+05,  // 0x412457D27719AD5C
346 
        -2.94490264303834643215e+05, // 0xC111F9690EA5AA18
347 
}
348 
 
349 
// for x in [8,4.5454] = 1/[0.125,0.22001]
350 
var q1R5 = [6]float64{
351 
        -2.08979931141764104297e-11, // 0xBDB6FA431AA1A098
352 
        -1.02539050241375426231e-01, // 0xBFBA3FFFCB597FEF
353 
        -8.05644828123936029840e+00, // 0xC0201CE6CA03AD4B
354 
        -1.83669607474888380239e+02, // 0xC066F56D6CA7B9B0
355 
        -1.37319376065508163265e+03, // 0xC09574C66931734F
356 
        -2.61244440453215656817e+03, // 0xC0A468E388FDA79D
357 
}
358 
var q1S5 = [6]float64{
359 
        8.12765501384335777857e+01,  // 0x405451B2FF5A11B2
360 
        1.99179873460485964642e+03,  // 0x409F1F31E77BF839
361 
        1.74684851924908907677e+04,  // 0x40D10F1F0D64CE29
362 
        4.98514270910352279316e+04,  // 0x40E8576DAABAD197
363 
        2.79480751638918118260e+04,  // 0x40DB4B04CF7C364B
364 
        -4.71918354795128470869e+03, // 0xC0B26F2EFCFFA004
365 
}
366 
 
367 
// for x in [4.5454,2.8571] = 1/[0.2199,0.35001] ???
368 
var q1R3 = [6]float64{
369 
        -5.07831226461766561369e-09, // 0xBE35CFA9D38FC84F
370 
        -1.02537829820837089745e-01, // 0xBFBA3FEB51AEED54
371 
        -4.61011581139473403113e+00, // 0xC01270C23302D9FF
372 
        -5.78472216562783643212e+01, // 0xC04CEC71C25D16DA
373 
        -2.28244540737631695038e+02, // 0xC06C87D34718D55F
374 
        -2.19210128478909325622e+02, // 0xC06B66B95F5C1BF6
375 
}
376 
var q1S3 = [6]float64{
377 
        4.76651550323729509273e+01,  // 0x4047D523CCD367E4
378 
        6.73865112676699709482e+02,  // 0x40850EEBC031EE3E
379 
        3.38015286679526343505e+03,  // 0x40AA684E448E7C9A
380 
        5.54772909720722782367e+03,  // 0x40B5ABBAA61D54A6
381 
        1.90311919338810798763e+03,  // 0x409DBC7A0DD4DF4B
382 
        -1.35201191444307340817e+02, // 0xC060E670290A311F
383 
}
384 
 
385 
// for x in [2.8570,2] = 1/[0.3499,0.5]
386 
var q1R2 = [6]float64{
387 
        -1.78381727510958865572e-07, // 0xBE87F12644C626D2
388 
        -1.02517042607985553460e-01, // 0xBFBA3E8E9148B010
389 
        -2.75220568278187460720e+00, // 0xC006048469BB4EDA
390 
        -1.96636162643703720221e+01, // 0xC033A9E2C168907F
391 
        -4.23253133372830490089e+01, // 0xC04529A3DE104AAA
392 
        -2.13719211703704061733e+01, // 0xC0355F3639CF6E52
393 
}
394 
var q1S2 = [6]float64{
395 
        2.95333629060523854548e+01,  // 0x403D888A78AE64FF
396 
        2.52981549982190529136e+02,  // 0x406F9F68DB821CBA
397 
        7.57502834868645436472e+02,  // 0x4087AC05CE49A0F7
398 
        7.39393205320467245656e+02,  // 0x40871B2548D4C029
399 
        1.55949003336666123687e+02,  // 0x40637E5E3C3ED8D4
400 
        -4.95949898822628210127e+00, // 0xC013D686E71BE86B
401 
}
402 
 
403 
func qone(x float64) float64 {
404 
        var p, q [6]float64
405 
        if x >= 8 {
406 
                p = q1R8
407 
                q = q1S8
408 
        } else if x >= 4.5454 {
409 
                p = q1R5
410 
                q = q1S5
411 
        } else if x >= 2.8571 {
412 
                p = q1R3
413 
                q = q1S3
414 
        } else if x >= 2 {
415 
                p = q1R2
416 
                q = q1S2
417 
        }
418 
        z := 1 / (x * x)
419 
        r := p[0] + z*(p[1]+z*(p[2]+z*(p[3]+z*(p[4]+z*p[5]))))
420 
        s := 1 + z*(q[0]+z*(q[1]+z*(q[2]+z*(q[3]+z*(q[4]+z*q[5])))))
421 
        return (0.375 + r/s) / x
422 
}

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